Analytic Methods in Birational Geometry

Plurisubharmonic (psh) functions can be viewed as a complex analytic analogue of convex functions, but they exhibit a much broader range of possible singularities, including those induced by ideals of holomorphic functions. Understanding psh singularities is a key issue in several questions of complex geometry, as illustrated by recent progress on the Yau-Tian-Donaldson conjecture. A powerful tool to do so is the classical notion of Lelong numbers and the theory of multiplier ideals, which open the way to a valuative approach that studies the analogue of psh functions on a (non-Archimedean) space of valuations. This lecture will present a gentle introduction to this circle of ideas.

In the last 50 years pluripotential theory has played a central role in order to solve geometric problems, such as the existence of special metrics (e.g. Kähler-Einstein, csck) on a compact Kähler manifold. In this talk I am going to present some recent developments in pluripotential theory. These new tools are so flexible that they allow to study “singular” settings : we will then be able to work with a singular variety and/or to search for singular metrics. The talk is based on a series of joint papers with Támas Darvas and Chinh Lu.

(Talk accessible for researchers in geometry).

Let X be a complex projective manifold, and let Y be a prime divisor in X. If Y is ample, it is well-known that the complement X\Y is an affine variety. Vice versa, assume that X\Y is affine, or more generally a Stein manifold. Then X\Y does not contain any curve, in particular Y has positive intersection with every curve. This leads to our main question: if X\Y is Stein, what can we say about the normal bundle of Y? After some general considerations I will focus on the case where Y is the projectivised tangent bundle of some manifold M, and X is a “canonical extension”. We will see that the Stein property leads to many restrictions on the birational geometry of M. This is work in progress with Thomas Peternell.

Let K(n,V) be the space of n-dimensional compact Kähler-Einstein manifolds with negative scalar curvature and volume bounded above by V. We prove that any sequence in K(n, V) converges in pointed Gromov-Hausdorff topology to a finite union of complete Kähler-Einstein metric spaces without loss of volume, which is biholomorphic to an algebraic semi-log canonical model with its non-log terminal locus removed. We further show that the Weil-Petersson metric extends uniquely to a Kähler current with continuous local potentials on the KSB compactification of the moduli space of canonically polarized manifolds. In particular, the Weil-Petersson volume of the KSB moduli space is finite.